Complex integral

How would I go about solving the line integral of ((e^z)+z)/(z-2))dz along the unit circle and along a circle of radius 3 centered at 0? I know how to parametrize the curves, so that's not a problem. I tried using the standard method for complex line integrals, but it got really messy, so I assume there is an easier way. I suspect I'm supposed to use Cauchy's formula, but I haven't had much practice with it and I'm not sure how to apply it here.

How would I go about solving the line integral of ((e^z)+z)/(z-2))dz along the unit circle and along a circle of radius 3 centered at 0? I know how to parametrize the curves, so that's not a problem. I tried using the standard method for complex line integrals, but it got really messy, so I assume there is an easier way. I suspect I'm supposed to use Cauchy's formula, but I haven't had much practice with it and I'm not sure how to apply it here.

Hint: Where are the poles for

For example:
When you integrate over a closed curve the result is proportional to the residue of the pole. If there is no pole inside the closed curve then the integral is 0. Thus your integration over the unit circle will be 0.